Pointy meshes pierce cloth sim. How can I get around this while also keeping the physical qualities of my cloth (e.g. mass, stiffness, etc.)?
As far as I can see, there are two issues causing this problem.

Keep in mind I am using an early 2015 MacBook air to do this so I cannot go to as high samples as you or others reading this can.


Blender calculates most of its collisions through vertices. As they come within close proximity of one another, it slows the vertices velocity. If it is moving fast or there are more vertices pulling it down (As in your mesh), blender has a tenancy to let the vertex pass. So to fix this we need to increase the number of collision vertices. This also can be done using two changes. One is to up the number subdivisions of the cloth mesh and the other method is to up the number of vertices of the point. 


Subdividing the mesh is simple enough. Hit subdivide. But to get performance, you may need to select the area close to the point and subdivide that a lot, as I have done in my image below.

Adding more vertices on the point. This can be done by adding a sphere or icosphere to the point of the object. This needs to be small enough that it won't be noticeable from the distance you are rendering at. If it is far, it can be quite large, if it is close, you will need to make it smaller. I have made an example, though it is of low resolution, it still fixes the problem. If you don't want the ball in the render, turn it off in the Outliner window.




The quality steps and collision steps. They will only do so much to the quality of your simulation, but it does help. In my final output, I set my quality steps to 25 and collision steps to 5. The one that made the most difference was the quality, but I think both should be increased.


Here is my result:



Remember, this is on a low power laptop. It took around 1 min to produce the first 58 frames. Yours may be at the same by the time you up the quality. In my render, you can see the change in mesh in the final result between High subdivisions and low subdivisions. If you want to minimise that effect, subdivide on an ever-shrinking area so it fades into the higher quality point of contact.

Hope I helped and best of luck with your project,

BFB